Converting Fractions from Binary to Decimal

Date: 02/25/2002 at 09:36:31
From: Jay
Subject: Converting Binary Fractions to Decimal 

Can you explain how to convert binary fractions to decimal numbers, 
e.g. 0.00011001100110011001...?

Thank you.

Date: 02/26/2002 at 00:18:37
From: Doctor Twe
Subject: Re: Converting Binary Fractions to Decimal 

Hi Jay - thanks for writing to Dr. Math.

Check out our FAQ on Number Bases at:

   http://mathforum.org/dr.math/faq/faq.bases.html   

it has a lot of information on converting both integers and fractions 
to other bases. But I didn't see an example of converting a fraction 
FROM binary back TO decimal, so here are the methods and an example.

The place values to the right of the radix point (what we call the 
equivalent of the "decimal point" when working in other bases) are 
simply negative powers of two:

     [.] 2^(-1)   2^(-2)   2^(3)   2^(-4)   ...

When a base is raised to a negative power, it just means the 
reciprocal of the base raised to that (positive) power. For example, 
2^(-1) = 1/(2^1) = 1/2, and 2^(-2) = 1/(2^2) = 4, etc. So the place 
values can also be expressed as:

     [.] 1/2   1/4   1/8   1/16   ...

Of course, we can also express these fractions in decimal form as:

     [.] .5   .25   .125   .0625   ...

So the binary fraction 0.1101 represents:

       1/2   1/4   1/8   1/16
       ---   ---   ---   ----
     .  1     1     0     1

Just as with integers, we multiply each digit by its place value and 
add the results. For my example, we'd get:

       1 * 1/2  = 1/2  =  8/16          1 * .5    = .5
     + 0 * 1/4  = 1/4  =  0/16          1 * .25   = .25
     + 1 * 1/8  =  0   =  2/16   or   + 0 * .125  = .0
     + 1 * 1/16 = 1/16 =  1/16          1 * .0625 = .0625
                         -----                      -----
                         11/16                      .8125

As an alternate method, you can simply move the radix point to the 
right end of the number (as long as it's not a repeating fraction). 
Count how many places you moved the radix point. Convert the resulting 
integer to binary, and divide by 2^n, where n is the number of places 
you moved the radix point.

In my example, we can convert .1101 to decimal by:

                                                   .1101

     1. Shift the radix point to the right end     1101.

     2. Count the number of places moved           (4 places)

     3. Convert the integer to decimal             1101_2 = 13_10

     4. Divide the result by 2^n                   13/(2^4) =
                                                   13/16    =
                                                   .8125

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.com/dr.math/